On the Maximum Spread of Non-Negative Matrices

Given a directed graph $G$, the spread of $G$ is the largest distance between any two eigenvalues of its adjacency matrix. In 2022, Breen, Riasanovsky, Tait, and Urschel asked what $n$-vertex directed graph maximizes spread, and whether this graph is undirected. We prove the more general result that the spread of any $n\times n$ non-negative matrix $A$ with $\mathrm{\parallel }A{\mathrm{\parallel }}_{\max }\le 1$ is at most $2n\mathrm{/}\sqrt{3}$, which is tight up to an additive factor and exact when $n$ is a multiple of three. Furthermore, our results show that the matrix with maximum spread is always symmetric.